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Journal of Theoretical Biology 245 (2007) 44–58

Regular Article

Markov chain approach to analyze the dynamics of pathogen

fecal shedding—Example of Listeria monocytogenes shedding in

a herd of dairy cattle

Renata Ivaneka,?, Yrjo ¨ T. Gro ¨ hna, Alphina Jui-Jung Hob, Martin Wiedmannb

aDepartment of Population Medicine and Diagnostic Sciences, College of Veterinary Medicine, Cornell University, Ithaca, NY 14853, USA

bDepartment of Food Science, 412 Stocking Hall, Cornell University, Ithaca, NY 14853, USA

Received 1 March 2006; received in revised form 25 July 2006; accepted 27 September 2006

Available online 7 October 2006

Abstract

Fecal shedding is an important mechanism of spreading of a number of human and animal pathogens. Understanding of the dynamics

of pathogen fecal shedding is critical to be able to control or prevent the spread of diseases caused by these pathogens. The objective of

this study was to develop a model for analysis of the dynamics of pathogen fecal shedding. Fecal shedding of Listeria monocytogenes in

dairy cattle was used as a model system. A Markov chain model (MCM) with two states, shedding and non-shedding, has been developed

for overall L. monocytogenes fecal shedding (all L. monocytogenes subtypes) and fecal shedding of three L. monocytogenes subtypes

(ribotypes 1058A, 1039E and 1042B) using data from one study farm. The matrices of conditional probabilities of transition between

shedding and non-shedding states for different sets of covariates have been estimated by application of logistic regression. The covariate-

specific matrices of conditional probabilities, describing the presence of different risk factors, were used to estimate (i) the stationary

prevalence of dairy cows that shed any L. monocytogenes subtype or ribotypes 1058A, 1039E, and 1042B, (ii) the duration of overall and

subtype specific fecal shedding, and (iii) the duration of periods without shedding. A non-homogeneous MCM was constructed to study

how the prevalence of fecal shedders changes over time. The model was validated with data from the study farm and published literature.

The results of our modeling work indicated that (i) the prevalence of L. monocytogenes fecal shedders varies over time and can be higher

than 90%, (ii) L. monocytogenes subtypes exhibit different dynamics of fecal shedding, (iii) the dynamics of L. monocytogenes fecal

shedding are highly associated with contamination of silage (fermented feed) and cows’ exposure to stress, and (iv) the developed

approach can be readily used to study the dynamics of fecal shedding in other pathogen–host–environment systems.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Markov chain model; Logistic regression; Duration of fecal shedding; Silage; Stress

1. Introduction

Fecal shedding represents an important mechanism of

spreading of a number of pathogens. Estimating the

probability and the duration of pathogen fecal shedding

and the influence of risk factors on that probability and

duration are of fundamental importance in the control of

diseases that use fecal shedding as a means of spreading.

Pathogen fecal shedding of an individual may change

over time with interchanging episodes of shedding and

non-shedding. This two-state (shedding and non-shedding

states) dynamic process is also often accompanied by

dynamic occurrence of risk factors (covariates) associated

with fecal shedding, forming a complex two-state-covariate

dynamic structure. Because of that, the analysis of

pathogen fecal shedding requires an analytical tool capable

of handling the multi-state and dynamic nature of fecal

shedding and the dynamic nature of risk factors associated

with it. Furthermore, the method should be able to predict

the pattern of fecal shedding beyond the range and

timeframe of available data. A variety of descriptive

ARTICLE IN PRESS

www.elsevier.com/locate/yjtbi

0022-5193/$-see front matter r 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jtbi.2006.09.031

?Corresponding author. Tel.: +16072533052; fax: +16072533083.

E-mail addresses: ri25@cornell.edu (R. Ivanek), ytg1@cornell.edu

(Y.T. Gro ¨ hn), ajh22@cornell.edu (A. Jui-Jung Ho), mw16@cornell.edu

(M. Wiedmann).

Page 2

(statistical) techniques has been used to analyze fecal

shedding data, many borrowed from survival analysis (e.g.

Alali et al., 2004). Within the framework of survival

analysis, the recurrent events analysis (Therneau and

Grambsch, 2000) could be used to analyze the dynamic

and intermittent nature of fecal shedding. However,

because the time spent in a fecal shedding episode could

be long relative to the time spent between shedding

episodes, a recurrent events analysis may not be appro-

priate. Instead, a more suitable approach would involve

development of multi-state models, where the focus is on

the times spent in various states of shedding, and on

transition rates between the states (McKay et al., 2006).

Whether survival analysis is suitable for analysis of the

multi-state and dynamic nature of fecal shedding or not, a

general disadvantage of all descriptive techniques (includ-

ing survival analysis) is their applicability to only the data

set from which statistics have been calculated. Because of

that there are no grounds other than expert opinion or

experimental confirmation with which to determine if they

can be used to assess the relationships in untested

circumstances, i.e. in some other system (location/popula-

tion), or even in the same system at another time (WHO,

2005). A group of models that generally allows predictions

beyond untested circumstances are known as mechanistic

(also known as explanatory and process) models. They are

built by explicitly considering the processes that produce

our observations (Ellner and Guckenheimer, 2006) and

therefore could expand the range of the data to include any

untested circumstances where the same process can be

presumed to operate, i.e. in situations where no measured

data are available or even where such measurements are

impossible (WHO, 2005). A mechanistic approach suitable

for analysis of systems whose behavioral characteristics

involve multi-state and dynamic elements, such as fecal

shedding data, is Markov chain (MC) analysis. An MC

model is characterized by two components: states and

transition probabilities, which are synonyms for variables

and parameters. The probability of the transition from the

current state of a MC model to another state is solely

determined by the current state.

The objective of this study was to apply MC modeling to

analyze the dynamics of pathogen fecal shedding, particu-

larly to estimate the duration of pathogen fecal shedding

and the prevalence of individuals that excrete the pathogen

in feces under the presence of different risk factors. Fecal

shedding of Listeria monocytogenes (LM) in a herd of dairy

cattle was used as a model system. This pathogen–hos-

t–environment system was chosen because of convenience

of gathering data (LM is commonly isolated from feces of

dairy cattle (Nightingale et al., 2004)) and relevance to

human health (cattle farms have been suggested as an

important reservoir of human LM infections (Arimi et al.,

1997; Nightingale et al., 2004)). Isolation of LM in feces of

cattle may be indicative of clinically or subclinically

infected animals (Gronstol, 1979; Loken et al., 1982;

Wesley, 1999). In addition, LM from contaminated feed

may pass through an animal’s digestive tract, without

causing LM infection, and also be excreted through feces

(Shepherd et al., 2000). While there is a strong indication

that LM infection occurs as a consequence of consumption

of LM in contaminated silage (Fenlon et al., 1996), it is

unknown whether LM excreted by a cow could cause LM

infection or passive shedding in other cows in the herd. A

cow-to-cow transmission of LM infection seems unlikely

because, even if LM is present at high levels in fecal

material, a cow would likely have to ingest a considerable

amount of fecal material to become infected (reviewed by

Ivanek et al., 2006). It has been reported that stress, such as

during transport, increases shedding of LM in feces

(Fenlon et al., 1996). The duration of fecal shedding

(either infected or pass through shedding) is not well

understood. Similarly, it is unclear whether LM infection in

cattle triggers immunity to new LM infections and how

long this immunity may persist (reviewed by Ivanek et al.,

2006).

To develop an MC model of LM fecal shedding, first we

applied statistical tools (univariate analysis and logistic

regression) to analyze how fecal shedding depends on time-

dependent covariates (covariates whose values are subject

to change with time), also referred to as predictor variables

or risk factors. Next, the estimated probability of fecal

shedding for a given set of covariates was translated into an

appropriate transition probability of a homogeneous

Markov chain (HMC) model to estimate the proportion

of dairy cattle likely to shed LM in feces and the duration

of LM fecal shedding. Finally, the covariate-specific HMC

models were combined into a non-homogeneous Markov

chain (N-HMC) model where the transition matrix of

conditional probabilities varies over time depending on the

presence of time-varying covariates, to study the day-to-

day variability of the proportion of cattle likely to shed LM

in feces. Before we proceed to the description of the

developed modeling approach we would like to make clear

that the purpose of the applied modeling was not to

analyze the transmission of diseases that use fecal shedding

as a means of spreading. Instead, the purpose was to

analyze the dynamics of fecal shedding (dynamics of

contagiousness) during infection or passive shedding,

whose understanding could then complement development

of models of disease transmission.

2. Materials and methods

2.1. Description of data

The data from an intensive longitudinal study of LM

shedding and feed contamination previously described by

Ho et al. (2006) were used. Briefly, in that study, a set of 25

randomly selected lactating dairy cattle from a herd of 52

cows, housed in a tie-stall barn, were used for daily

collection of fecal samples from January 18–31, 2004

(denoted as days 1–14), from February 12–15, 2004

(denoted as days 26–29), on February 25, 2004 (denoted

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as day 39) and from April 12–25, 2004 (denoted as days

86–99). Fecal sample aliquots of 10g were tested for LM

using an enrichment protocol. Cows that were dried off

(ceased producing milk) or sold in the intervals between the

sampling periods were replaced with the cow occupying her

former stall. A total of 32 different cows was sampled over

the study interval. During the same period (with the

exception of 1 day, February 25), two samples of the silage

used to feed the study herd were collected daily. In

addition, two silage samples were collected on January

17, 2004 (denoted as day 0). If LM was isolated from either

of the two samples, all silage was considered contaminated

for that day. This implies that all cows in the herd ate silage

of the same contamination status. Ribotyping (Wiedmann

et al., 1996) was performed to identify LM subtypes in

feces and silage samples. Generally, only one isolate

was ribotyped per sample of silage and feces. The

implications of ribotyping only one isolate per sample are

discussed under ‘‘The dynamics of LM fecal shedding are

highly associated with contamination of silage’’ in the

Section 4. On two occasions (January 26 and February 13),

all cows were de-liced with Atroban [with permethrin

(synthetic pyrethroid) as one of the active ingredients

(Anonymous, 2006)]. Ho et al. (2006) speculated that the

stress due to the antiparasitic treatment or the medication

itself could have had an immunosuppressive effect on cows.

Their hypothesis is supported by the fact that cyperme-

thrin, a pyrethroid similar to permethrin, which is one of

the active ingredients in Atroban, has previously been

shown to cause immunosuppression in goats (Tamang

et al., 1988).

For each of 25 cows, whose fecal sample was analyzed

daily for the presence of LM, we recorded whether LM was

isolated from feces (any subtype) and which LM subtype

was isolated. This information (presence of any LM and

LM subtype isolated) represented the response variables

for overall and subtype specific LM shedding at time t,

respectively. In addition, we noted whether any or the same

LM ribotype was isolated from feces of the same cow (i) a

day earlier (at time t?1) and (ii) 2 days earlier (at time

t?2); whether any or the same LM ribotype was isolated

from silage (i) on the same day that the feces sample was

collected (at time t), (ii) a day before fecal sample collection

(at time t?1), and (ii) two days before fecal sample

collection (at time t?2); and exposure to stress in the form

of antiparasitic treatment (i) on the same day that the feces

sample was collected (at time t) and (ii) a day before

collection of the fecal sample (at time t?1). All fecal

shedding, silage contamination and stress exposure data

were represented as binary variables indicating whether or

not LM or a particular LM subtype was isolated from a

particular fecal sample or silage sample and whether a cow

was exposed to stress, respectively. They were coded 1 if

LM or stress was present and 0 if absent. Fecal shedding

data for cows that were lost from follow up or enrolled

after the beginning of the study were coded as missing

observations.

There were 825 observations for fecal shedding at time t.

Silage samples were not collected on 1 day in the study

period. Therefore, there were 800 observations for feed

contamination at time t. Recording the shedding state and

feed contamination status a day and 2 days earlier resulted

in the loss of observations from 3 to 7 days, respectively.

There were 750 observations for fecal shedding and feed

contamination at time t?1 and 650 for fecal shedding and

feed contamination at time t?2.

2.2. Statistical analysis

Statistical analysis was performed in STATA (Inter-

cooled Stata, Version 6.0, Stata Corporation, College

Station, TX). A 2-stage process was used to model the

probability of overall and subtype specific LM fecal

shedding at time t. In the first stage, w2tests were done to

relate the probability of LM fecal shedding to each

explanatory variable. Only explanatory variables asso-

ciated (pp0.05) with the probability of shedding in these

unconditional w2tests advanced to the second stage,

logistic regression modeling. Fisher’s exact test was used

when one or more of the expected cell frequencies in 2?2

tables were less than 5. Although univariate analysis was

performed for all LM subtypes isolated from feces of cattle

in the study by Ho et al. (2006), only LM subtypes for

which we detected a significant association between the

response variable (fecal shedding at time t) and fecal

shedding of the same subtype at t?1 and/or t?2 and at

least one other explanatory variable were taken into the

next stage of statistical analysis, the logistic regression, and

subsequently to MC modeling. The association with at

least one other explanatory variable was not a necessary

condition to fit a MC model but it allowed us to

demonstrate modeling of the influence of time-varying

covariate(s) on fecal shedding. LM subtypes carried into

the subsequent analysis were ribotypes 1042B, 1058A and

1039E.

Logistic regression modeling: Logistic regression models

were used to model the effect of several explanatory

variables on the probability of overall and subtype specific

LM fecal shedding at time t. Eqs. (1)–(4) show the final

models for overall LM fecal shedding and fecal shedding of

ribotypes 1058A, 1039E and 1042B, respectively:

?

þ p½Stressðt ? 1Þ?,

?

þ p½S1058Aðt ? 1Þ?,

?

þ p½S1039Eðt ? 1Þ? þ p½Stressðt ? 1Þ?,

Log

p½FallðtÞ?

1 ? p½FallðtÞ?

?

¼ p½Fallðt ? 1Þ? þ p½Sallðt ? 1Þ?

ð1Þ

Log

p½F1058AðtÞ?

1 ? p½F1058AðtÞ?

?

¼ p½F1058Aðt ? 1Þ? þ p½S1058AðtÞ?

ð2Þ

Log

p½F1039EðtÞ?

1 ? p½F1039EðtÞ?

?

¼ p½F1039Eðt ? 1Þ?

ð3Þ

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Log

p½F1042BðtÞ?

1 ? p½F1042BðtÞ?

??

¼ p½F1042Bðt ? 1Þ? þ p½S1042BðtÞ?.

(4)

Theterms

p[Fall(t)],

p[F1058A(t)],

p[F1039E(t)],and

p[F1042B(t)] represent the probabilities that a fecal sample

is positive at time t for any LM subtype, and ribotypes

1058A, 1039E and 1042B, respectively. Similarly, p[Fall

(t?1)], p[F1058A(t?1)], p[F1039E(t?1)], and p[F1042B(t?1)]

represent the probabilities that a fecal sample is positive

at time t?1 for any LM subtype, and ribotypes 1058A,

1039E and 1042B, respectively. The terms p[Sall(t?1)],

p[S1058A(t?1)], and p[S1039E(t?1)] denote the probabilities

of silage contamination with any LM subtype, and

ribotypes 1058A, and 1039E at time t?1, respectively.

Similarly, p[S1058A(t)] and p[S1042B(t)] are the probabilities

of silage contamination at time t with ribotypes 1058A and

1042B, respectively. Finally, p[Stress(t?1)] is the prob-

ability that the cow was exposed to stress at time t?1. To

select final models, a stepwise backward variable selection

approach was used based on the likelihood-ratio statistic,

and removal probabilities of pp0.05.

Probability of LM fecal shedding: The probabilities of

overall LM fecal shedding and shedding of ribotypes

1058A, 1039E, and 1042B at time t were estimated by Eqs.

(5)–(8) from parameter estimates calculated from the

logistic regression models (Eqs. (1)–(4)). The parameter

estimates (in Eqs. (5)–(8) denoted as a, b, c and d, with

numbers in subscripts representing different explanatory

variables) represent the difference in the log odds of overall

and subtype specific LM fecal shedding at time t between

levels of explanatory variables:

p½FallðtÞ?

¼

1

1 þ exp ?ða0þ a1Sallðt ? 1Þ þ a2Fallðt ? 1Þ þ a3Stressðt ? 1ÞÞðÞ,

ð5Þ

p½F1058AðtÞ?

¼

1 þ exp ? b0þ b1S1058AðtÞ þ b2S1058Aðt ? 1Þ þ b3F1058Aðt ? 1Þ

1

ðÞ

ð6Þ

ðÞ,

p½F1039EðtÞ?

¼

1 þ exp ? c0þ c1S1039Eðt ? 1Þ þ c2F1039Eðt ? 1Þ þ c3Stressðt ? 1Þ

1

ðÞ

ð7Þ

ðÞ,

p½F1042BðtÞ?

¼

1

1 þ exp ?ðd0þ d1S1042BðtÞ þ d2F1042Bðt ? 1ÞÞðÞ.

ð8Þ

2.3. Markov chain modeling

MC modeling was performed in MATLAB 6.5 (Math-

Works, Inc., Natick, MA). Covariate-specific MC models

with two states, LM shedding and LM non-shedding states,

were developed to analyze the dynamics of overall and

subtype specific fecal shedding. Transition probabilities of

these MC models were estimated from probabilities of LM

fecal shedding for sets of covariates. Each covariate-

dependent matrix of Markov transition probabilities was

used to construct the corresponding HMC model in which

the transition matrix is the same at all time points. The

covariate-specific HMC models were combined into an N-

HMC model where the transition matrix of conditional

probabilities varies over time depending on the presence of

time-varying covariates. The theory explaining MC model-

ing applied here can be found in Lindsey (2004).

In fitting MC models we assumed that (1) pathogen fecal

shedding of a cow depends only on the cow’s pathogen

fecal shedding on a day earlier, and on contamination of

consumed silage and exposure to stress on the same day

and/or a day earlier depending on the model; (2) the study

farm used for model parameterization is representative of a

typical dairy farm, and (3) the detected fecal shedding,

contamination of silage and exposure to stress in the study

herd are real, implying perfect sensitivity and specificity of

tests used to detect LM in feces and silage and perfect

perceptionindetectingcows’

The implications of these assumptions are discussed in

Section 4.

Matrices of transition probabilities: Each developed

covariate-specific HMC model had an associated matrix

of transition probabilities in which the cell probabilities

sum to unity across rows. For this 2?2 table we defined

four terms, where Ftand Ft?1denote whether or not LM

was isolated from feces of a cow at times t and t?1,

respectively: p00¼ P(Ft¼ 0|Ft?1¼ 0), p01¼ P(Ft¼ 1|Ft?1

¼ 0), p10¼ P(Ft¼ 0|Ft?1¼ 1) and p11¼ P(Ft¼ 1|Ft?1¼

1). The transition probabilities p01and p11were estimated

by logistic regression modeling. Then, p00¼ (1?p01) and

p10¼ (1?p11).

Marginal and stationary distributions of HMC model:

Covariate dependent matrices of transition probabilities

were used to estimate the marginal distribution of cows at

time t in shedding and non-shedding states (Pt) for each set

of covariates. The marginal distribution gives the propor-

tion (probability) of cows in shedding and non-shedding

states at time t. The marginal probability of being in the

shedding state at time t (P1,t) and the marginal probability

of being in the non-shedding state at time t (P0,t) were

estimated as follows:

exposuretostress.

P1;t¼ p01P0;t?1þ p11P1;t?1,(9)

P0;t¼ p00P0;t?1þ p10P1;t?1.

The initial probabilities of being in the non-shedding and

shedding states when the Markov process starts, i.e. at time

t ¼ 0, were set to 1 and 0, respectively, meaning that when

the process starts all cows are in the non-shedding state.

After an MC process has run for a sufficiently long time the

marginal probabilities of being in the non-shedding and

(10)

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shedding states converged into a stationary marginal

distribution. The stationary distribution gives the equili-

brium probabilities (also referred to as the prevalences and

proportions) of being in non-shedding and shedding states.

It can also be interpreted as the long-run proportion of

time spent in each state. In addition to numerical

estimation, the stationary probabilities of being in the

non-shedding (P0,s) and shedding (P1,s) states were

determined analytically from the matrix of transition

probabilities by application of Eqs. (11) and (12):

P0;s¼ p10=ðp01þ p10Þ,

P1;s¼ p01=ðp01þ p10Þ.

Time spent in a state: The time spent in a state of an

HMC model has a geometric distribution, allowing us to

estimate the mean and variance of the residence time in the

non-shedding and shedding states. The mean and variance

of the duration of time spent in the non-shedding state

(E(T0) and Var(T0), respectively) and the shedding state

(E(T1) and Var(T1), respectively) were calculated as

follows:

(11)

(12)

EðT0Þ ¼ 1=ð1 ? p00Þ,

VarðT0Þ ¼ p00=ð1 ? p00Þ2,

EðT1Þ ¼ 1=ð1 ? p11Þ,

VarðT1Þ ¼ p11=ð1 ? p11Þ2.

N-HMC model of overall and subtype specific LM fecal

shedding: The covariate-specific HMC model developed for

overall and subtype specific LM fecal shedding were

combined into an N-HMC model where the transition

matrix of conditional probabilities of each state, given the

previous state, varies over time depending on the presence

of time-varying covariates. The feature of interest in the N-

HMC model is the marginal distribution. That is because

the matrix of transition probabilities of an N-HMC model

changes over time and a stationary distribution cannot be

reached (unless an N-HMC model reaches a limiting

distribution if the matrices of transition probabilities

converge to some constant matrix as t goes to infinity).

The N-HMC model could be run against any assumed or

real pattern of time-varying covariates (e.g. presence of

contaminated feed, exposure to stress).

Method application to other data sets: The method

developed here was applied to published time series data

on fecal shedding (Grif et al., 2003; Robinson et al., 2004)

to demonstrate its usefulness in the analysis of the

dynamics of fecal shedding in other pathogen–host–envir-

onment systems.

Model validation: While the best data for model

validation would be from a similar longitudinal study of

fecal shedding, stress and feed contamination obtained on

other dairy farms, to our best knowledge such data do not

exist. Therefore, the model was validated by two other

approaches. First, the model estimates were compared with

(13)

(14)

(15)

(16)

the data observed on the study farm. Model estimates used

for validation in this way were (a) the daily prevalence of

fecal shedders and (b) average prevalence (and variance) of

fecal shedders over the study period. The second approach

involved comparison of the variability of daily prevalences

of fecal shedders predicted by the model with the

variability of prevalences reported in the literature.

3. Results

3.1. Statistical analysis

Univariate analysis: Univariate analysis was performed

to assess association between response variables (fecal

shedding of any LM subtype, and LM ribotypes 1058A,

1039E and 1042B at time t) and explanatory variables.

Table 1 shows cross-tabulation of data between overall LM

fecal shedding and shedding of LM ribotypes 1058A,

1039E and 1042B and significantly associated explanatory

variables.

Logistic regression: The results of logistic regression are

presented in Table 2. If silage was contaminated with any

LM subtype at time t, the odds that a cow shed any LM

subtype in feces the next day increased by a factor of 14.0,

when other variables were controlled. Similarly, contam-

ination of silage with ribotypes 1058A, 1039E and 1042B

increased the odds of fecal shedding of the same subtype on

the day when contaminated silage was consumed [1058A

(odds ratio (OR) ¼ 12.3) and 1042B (OR ¼ 5.6)] and the

day after [1058A (OR ¼ 6.2) and 1039E (OR ¼ 3.6)]. If the

cow was exposed to stress, the odds of shedding any LM

subtype and ribotype 1039E the next day increased by

factors of 5.3 and 4.0, respectively. The odds of shedding

any LM subtype also increased by a factor of 3.3 if the

same cow excreted LM in feces a day earlier. Similarly,

fecal shedding of ribotypes 1058A, 1039E and 1042B

increased the odds of shedding the same subtype the next

day by factors of 16.3, 6.2 and 3.7, respectively.

Probability of LM fecal shedding: Parameter estimates

from the logistic regression analysis were used to derive the

probabilities of fecal shedding given different combinations

of covariates (Fig. 1). The probability that a cow shed any

LM in feces could be as high as 91% after exposure to

stress and consumption of contaminated feed. The prob-

ability that a cow shed ribotype 1058A could be even

higher, 95%, if the cow sheds the same subtype a day

earlier and its feed was contaminated with LM on the day

of feces sample collection and a day earlier. The probability

of shedding ribotype 1039E went up to 71% if the cow shed

the same LM subtype, ate silage contaminated with the

same subtype and was exposed to stress a day earlier. The

expected probability of fecal shedding of ribotype 1042B

was lower than the probability of other tested subtypes.

The highest probability, 37%, was for a cow that excreted

the same subtype a day earlier and ate silage contaminated

with the same subtype on the day of fecal sample

collection.

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Table 1

Variables (with notations and codes) significantly associated with fecal shedding of any L. monocytogenes (LM) subtype and ribotypes 1058A, 1039E and

1042B in feces at time t

Variable description (notation)Feces at t negative Feces at t positiveTotal%

p-Value

Total% Total%

All subtypes

Any LM subtype in feces at t?1 (Fall(t?1))

0

1

Any LM subtype in feces at t?2 (Fall(t?2))

0

1

Any LM subtype in silage at t (Sall(t))

0

1

Any LM subtype in silage at t?1 (Sall(t?1))

0

1

Any LM subtype in silage at t?2 (Sall(t?2))

0

1

Exposure to stress at t (Stress(t))

0

1

Exposure to stress at t?1 (Stress(t?1))

0

1

475

72

86.8

13.2

100

128

43.9

56.1

575

200

74.2

25.8

o0.001

361

91

79.9

20.1

100

98

50.5

49.5

461

189

70.9

29.1

o0.001

354

202

63.7

36.3

4618.9

81.2

400

400

50.0

50.0

o0.001

198

333

190

63. 7

36.3

177.5

92.5

350

400

46. 7

53.3

o0.001

210

287

165

63.5

36.5

136.6

93.4

300

350

46.2

53.9

o0.001

185

520

50

91.2

8.8

255 100.0

0.0

775

50

93.9

6.1

o0.001

0

496

27

94.8

5.2

204

23

89.9

10.1

700

50

93.3

6. 7

0.012

Ribotype 1058A

LM ribotype 1058A in feces at t?1 (F1058A(t?1))

0

1

LM ribotype 1058A in feces at t?2 (F1058A(t?2))

0

1

LM ribotype 1058A in silage at t (S1058A(t))

0

1

LM ribotype 1058A in silage at t?1 (S1058A(t?1))

0

1

LM ribotype 1058A in silage at t?2 (S1058A(t?2))

0

1

685

18

97.4

2.6

29

18

61.7

38.3

714

36

95.2

4.8

o0.001

612

11

98.2

1.8

17

10

63.0

37.0

629

21

96.8

3.2

o0.001

724

22

97.1

3.0

27

27

50.0

50.0

751

49

93.9

6.1

o0.001

635

68

90.3

9.7

15

32

31.9

68.1

650

100

86.7

13.3

o0.001

590

33

94.7

5.3

18

9

66.7

33.3

608

42

93.5

6.5

o0.001

Ribotype 1039E

LM ribotype 1039E in feces at t?1 (F1039E(t?1))

0

1

LM ribotype 1039E in silage at t (S1039E(t))

0

1

LM ribotype 1039E in silage at t?1 (S1039E(t?1))

0

1

Exposure to stress at t?1 (Stress(t?1))

0

1

686

26

96.4

3.7

28

10

73.7

26.3

714

36

95.2

4.8

o0.001

720

42

94.5

5.5

31

7

81.6

18.4

751

49

93.9

6.1

0.001

628

84

88.2

11.8

22

16

57.9

42.1

650

100

86.7

13.3

o0.001

668

44

93.8

6.2

32

6

84.2

15.8

700

50

93.3

6.7

0.021

Ribotype 1042B

LM ribotype 1042B in feces at t?1 (F1042B(t?1))

0

1

LM ribotype 1042B in feces at t?2 (F1042B(t?2))

0

1

691

31

95.7

4.3

23

5

82.1

17.9

714

36

95.2

4.8

0.001

610

18

97.1

2.9

19

3

86.4

13.7

629

21

96.8

3.2

0.03

R. Ivanek et al. / Journal of Theoretical Biology 245 (2007) 44–58

49

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3.2. Markov chain modeling

Matrices of transition probabilities: The probabilities of

fecal shedding (Fig. 1) were used to derive covariate specific

matrices of conditional probabilities of transition of cows

between the non-shedding and shedding states. For

example, for overall fecal shedding (all subtypes) when

silage is not contaminated and cows are not exposed to

stress at time t?1, the transition probabilities p01and p11

equal 0.038 and 0.116, respectively. Then, transition

probabilities p00and p10 equal 0.962 and 0.884, respec-

tively. These matrices were used to model change in the

proportion of fecal shedders over time through the MC

process. Because logistic regression modeling indicated that

fecal shedding can be better explained by fecal shedding

observed one day earlier (at time t?1) than that observed 2

days earlier (at time t?2), all developed models were first-

order MC models.

ARTICLE IN PRESS

Table 1 (continued)

Variable description (notation)Feces at t negativeFeces at t positiveTotal%

p-Value

Total% Total%

LM ribotype 1042B in silage at t (S1042B(t))

0

1

LM ribotype 1042B in silage at t?1 (S1042B(t?1))

0

1

730

42

94.6

5.4

2175.0

25.0

751

49

93.9

6.1

o0.001

7

637

85

88.2

11.8

13

15

46.4

53.6

650

100

86.7

13.3

o0.001

Variables are coded 1 if LM was present and 0 if absent.

LM ¼ L. monocytogenes.

Table 2

Results of logistic regression modeling with notations used for parameters and parameter estimates

Parameter description (notation) Parameter estimate

(notation)

Odds ratioConfidence limits for odds ratio

Lower 5%Upper 95%

All subtypes

Probability of silage contamination with any LM subtype at time t?1

(p[Sall(t?1)])

Probability of fecal shedding of any LM subtype at time t?1 (p[Fall(t?1)])

Probability of exposure to stress at time t?1 (p[Stress(t?1)])

Intercept

2.64 (a1) 14.07.925.0

1.20 (a2)

1.66 (a3)

?3.22 (a0)

3.3

5.3

2.2

2.5

5.0

11.1

Ribotype 1058A

Probability of silage contamination with ribotype 1058A at time t

(p[S1058A(t)])

Probability of silage contamination with ribotype 1058A at time t?1

(p[S1058A(t?1)])

Probability of fecal shedding of ribotype 1058A at time t?1

(p[F1058A(t?1)])

Intercept

2.51 (b1)12.3 4.931.2

1.82 (b2) 6.22.8 13.8

2.79 (b3)16.36.3 42.3

?4.15 (b0)

Ribotype 1039E

Probability of silage contamination with ribotype 1039E at time t?1

(p[S1039E(t?1)])

Probability of fecal shedding of ribotype 1039E at time t?1

(p[F1039E(t?1)])

Probability of exposure to stress at time t?1 (p[Stress(t?1)])

Intercept

1.28 (c1)3.61.77.8

1.82 (c2)6.22.415.7

1.39 (c3)

?3.61 (c0)

4.01.510.6

Ribotype 1042B

Probability of silage contamination with ribotype 1042B at time t

(p[S1042B(t)])

Probability of fecal shedding of ribotype 1042B at time t?1

(p[F1042B(t?1)])

Intercept

1.72 (d1) 5.62.214.4

1.32 (d2)3.71.311.1

?3.59 (d0)

LM ¼ L. monocytogenes.

R. Ivanek et al. / Journal of Theoretical Biology 245 (2007) 44–58

50

Page 8

Marginal and stationary distributions of HMC models:

Covariate specific transition matrices were used to con-

struct covariate specific HMC models. In each of these

models we evaluated marginal and stationary distributions.

Fig. 2 shows the marginal proportion of cows that shed any

LM and ribotypes 1058A, 1039E and 1042B for different

sets of covariates. The initial distribution, the marginal

distribution when the process starts, of cows in the

shedding and non-shedding states were set to 0 and 1

respectively in all developed HMC models. Thereafter, Fig.

2 shows that the marginal proportion of shedders

increased, showing the underlying marginal trend of the

process. After a certain number of steps the marginal

proportion of shedders converged into a stationary

(equilibrium) prevalence. Each developed covariate-specific

HMC model was irreducible (every state is accessible from

every other state) and positive recurrent (the expected

return time is finite for every state), and as such it had a

unique stationary distribution. The stationary distribution

of all developed HMC models was insensitive to the initial

distribution. Table 3 shows the stationary prevalence of

cows in shedding and non-shedding states for different

combinations of covariates. The stationary proportion of

cows that shed any LM could be between 4% and 89%

depending on the contamination of silage and exposure to

stress. Similarly, the proportion of cows that shed ribotype

1039E could be between 3% and 49% also depending on

the contamination status of feed and exposure to stress.

The stationary proportion of cows that were likely to shed

ribotype 1058A could be between 2% and 92% depending

ARTICLE IN PRESS

Fig. 1. Probability of fecal shedding for different sets of covariates.

0

0.2

0.4

0.6

0.8

1

05 10

Days

Probability

Sall (t-1) = 0, Stress (t-1) = 0

Sall (t-1)=0, Stress (t-1) = 1

Sall (t-1) = 1, Stress (t-1) = 0

Sall (t-1)=1, Stress (t-1) = 1

0

0.2

0.4

0.6

0.8

1

05 10

Days

Probability

S1058A (t-1) = 0, S1058A (t) = 0

S1058A (t-1) = 0, S1058A (t) = 1

S1058A (t-1) = 1, S1058A (t) = 0

S1058A (t-1) = 1, S1058A (t) = 1

0

0.2

0.4

0.6

0.8

1

05 10

Days

Probability

S1039E (t-1) = 0, Stress (t-1) = 0

S1039E (t-1) = 1, Stress (t-1) = 0

S1039E (t-1) = 0, Stress (t-1) = 1

S1039E (t-1) = 1, Stress (t-1) = 1

0

0.2

0.4

0.6

0.8

1

0510

Days

Probability

S1042 (t) = 0

S1042 (t) = 1

(A)

(B)

(C)

(D)

Fig. 2. Marginal probability of fecal shedders of (A) any Listeria monocytogenes subtype and ribotypes (B) 1058A, (C) 1039E and (D) 1042B for different

combinations of covariates.

R. Ivanek et al. / Journal of Theoretical Biology 245 (2007) 44–58

51

Page 9

on the silage contamination only. The stationary distribu-

tion of cows shedding ribotype 1042B could be 3% or 17%

depending on the contamination of silage.

Time spent in a state: The conditional probabilities of

transition from the non-shedding state at time t?1 to the

non-shedding state at time t (p00) and from the shedding

state at time t?1 to the shedding state at time t (p11) were

used to estimate the mean and variance of the time spent in

thenon-sheddingandshedding

(Table 3). On average, a cow shed LM (any subtype) for

1–11 days, while episodes of non-shedding on average

lasted one to 26 days depending on silage contamination

and stress. Similarly, the mean duration of time during

which a cow was likely to shed ribotype 1039E was from

one to three days depending on the contamination status of

silage and exposure to stress. Non-shedding episodes of

this subtype could last from four to 38 days. Ribotype

1058A could be shed from one to 21 days depending on the

silage contamination. However, non-shedding episodes

could last from two to 64 days. In contrast to the mean

duration of overall LM shedding (all subtypes) and mean

duration of shedding of ribotypes 1058A and 1039E, the

mean duration of shedding of ribotype 1042B was short,

1–2 days. On the other hand, the mean duration of non-

shedding episodes of this subtype could be 7–37 days

depending on the contamination status of silage.

N-HMC model of overall and subtype specific LM fecal

shedding: The HMC models developed for each set of

covariates were combined into an N-HMC model where

the transition matrix of conditional probabilities of each

state, given the previous state, varies over time depending

on the presence of time-varying covariates (contamination

states, respectively

of feed, or exposure to stress, or both). An advantage of an

N-HMC model compared to multiple covariate-specific

HMC models is that the effects of the covariates are

allowed to change over time. The cost, however, is an

increased variability in the predicted marginal prevalence

of fecal shedders, unless the predicted variability is an

intrinsic characteristic of the system. The developed N-

HMC model could be run with any pattern of feed

contamination and exposure to stress. Here, we show the

result of the N-HMC model of overall LM fecal shedding

(all ribotypes) ran with the pattern observed on the study

farm (Table 4 and Fig. 3) because that setting also allowed

assessment of the model fit.

Model validation: The developed N-HMC models of fecal

shedding of any LM subtype and ribotypes 1058A, 1039E

and 1042B were run against the data on feed contamination

and exposure to stress observed on the study farm. Fig. 3

shows the daily prevalence of fecal shedders of any LM

subtype predicted by the N-HMC model plotted against the

ARTICLE IN PRESS

Table 3

Stationary probability of shedding of any L. monocytogenes subtype and ribotypes 1058A, 1039E and 1042B for different sets of covariates and respective

time spent in the non-shedding and shedding states

Combination of covariatesStationary distribution Time in non-shedding state (days)Time in shedding state (days)

MeanVariance MeanVariance

All subtypes

Sall(t?1) ¼ 0, Stress(t?1) ¼ 0

Sall(t?1) ¼ 0, Stress(t?1) ¼ 1

Sall(t?1) ¼ 1, Stress(t?1) ¼ 0

Sall(t?1) ¼ 1, Stress(t?1) ¼ 1

Ribotype 1058A

S1058A(t?1) ¼ 0, S1058A(t) ¼ 0

S1058A(t?1) ¼ 0, S1058A(t) ¼ 1

S1058A(t?1) ¼ 1, S1058A(t) ¼ 0

S1058A(t?1) ¼ 1, S1058A(t) ¼ 1

Ribotype 1039E

S1039E(t?1) ¼ 0, Stress(t?1) ¼ 0

S1039E(t?1) ¼ 0, Stress(t?1) ¼ 1

S1039E(t?1) ¼ 1, Stress(t?1) ¼ 0

S1039E(t?1) ¼ 1, Stress(t?1) ¼ 1

Ribotype 1042B

S1042(t) ¼ 0

S1042(t) ¼ 1

0.042

0.227

0.504

0.889

26.2

5.8

2.8

1.3

657.8

27.4

5.0

0.5

1.1

1.7

2.8

10.7

0.2

1.2

5.2

104.3

0.019

0.405

0.187

0.919

64.4

6.1

11.3

1.8

4080.8

31.6

115.4

1.3

4.2

2.6

20.7

0.3

13.3

4.1

405.71.5

0.03

0.141

0.124

0.489

38.0

10.2

11.3

3.6

1407.6

93.8

116.6

1.2

1.7

1.6

3.4

0.2

1.1

1.0

8.29.1

0.029

0.174

37.3

7.5

1351.4

48.6

1.1

1.6

0.1

0.9

Table 4

Overall prevalence of L. monocytogenes fecal shedders and the prevalences

of shedders of ribotypes 1058A, 1039E and 1042B estimated from the data

observed in the study herd and that predicted by the non-homogeneous

Markov chain model

ObservedModel prediction

MeanVarianceMean Variance

All subtypes

Ribotype 1058A

Ribotype 1039E

Ribotype 1049B

0.305

0.068

0.048

0.035

0.095

0.029

0.012

0.003

0.314

0.073

0.042

0.044

0.070

0.027

0.001

0.002

R. Ivanek et al. / Journal of Theoretical Biology 245 (2007) 44–58

52

Page 10

daily prevalence of fecal shedders observed in the study herd.

There were no data on covariates present on day 9 (February

25, 2004) of the longitudinal study. Therefore, that day was

excluded from validation. The plots of predicted daily

prevalence of fecal shedders of ribotypes 1058A, 1039E

and 1042B are not shown here. Table 4 shows the mean and

variance of the daily prevalence of fecal shedders estimated

from the observed data and those estimated from predictions

of the N-HMC model. The agreement between model

predictions and the observed data was good. To further

validate the developed modeling approach we compared the

variability of prevalences of fecal shedders predicted by

covariate specific HMC models with the published literature.

The predicted prevalences vary from 2% to 92%, which is

similar to the variability of within-herd prevalences of fecal

shedders, from a few percent to 100%, reported by Night-

ingale et al. (2004).

Application of the method to other pathogen–host–envir-

onment systems: The method developed here was applied to

the published data on daily LM fecal shedding observed in

healthy people over the course of one year (Grif et al.,

2003). The pattern of positive and negative fecal samples

was determined as accurately as possible from Fig. 1 in that

study. There was no information on time-varying covari-

ates. Our method estimated an average duration of LM

fecal shedding in healthy people and the duration of non-

sheddingepisodesto be

(Var ¼ 35,438) days, respectively. The estimated stationary

prevalence of healthy people that are likely to shed LM in

their feces was 1.2%.

2(Var ¼ 3)and 189

Our method for analysis of the dynamics of pathogen

fecal shedding was further applied to the published data on

fecal shedding of Escherichia coli O157 (Robinson et al.,

2004). The pattern of E. coli positive and negative fecal

samples was reconstructed from Fig. 5 in that study

showing individual shedding profiles observed in a cohort

of 16 naturally infected calves sampled once or twice a day

for 15 days (when more than one fecal sample was tested

per day, the result of a sample taken first was considered).

There was no information on the presence of explanatory

variables. An average duration of fecal shedding of E. coli

in calves and the duration of non-shedding episodes

estimated by our method were 8 (Var ¼ 63) and 2

(Var ¼ 3) days, respectively. The estimated stationary

prevalence of calves that shed E. coli in their feces was

79%.

4. Discussion

To study the dynamics of overall fecal shedding (all LM

subtypes) and fecal shedding of three subtypes (ribotypes

1058A, 1039E and 1042B) we applied MC analysis because

it is capable of handling the multi-state and dynamic nature

of fecal shedding and can describe prevailing shedding

states as well as future states in the intermittent pattern of

fecal shedding. The developed MC models allowed

estimation of the day-to-day and equilibrium proportions

of dairy cattle likely to shed LM in feces, and the duration

and frequency of LM fecal shedding depending on the

presence of time-varying covariates (here, contamination of

feed and exposure to stress).

In development of MC models of overall and subtype

specific fecal shedding in dairy cattle, several assumptions

have been made, whose possible effects on model estimates

should be discussed. We assumed that a cow’s pathogen

fecal shedding depends only on the cows’ pathogen fecal

shedding on a day earlier, and on (depending on the model)

contamination of consumed silage and/or exposure to

stress on the same day and/or a day earlier. This

assumption seems rational because it was based on the

results of statistical analysis, which preceded the develop-

ment of MC models. However, it is possible that fecal

shedding is influenced by other risk factors, which either

were not considered in the statistical analysis or were

considered but the study design did not have enough

statistical power to detect their association with fecal

shedding.

Another assumption that we have made in the model

development is that the study farm used for model

parameterization is representative of a typical dairy farm

and, consequently, that the estimates predicted from the

developed MC models are generalizable to the entire dairy

cattle population. While this assumption was necessary

because there are no other similar data on LM fecal

shedding in dairy cattle, it is hard to say how typical the

study farm really is. We can only say that, based on the

similarity between the range of prevalences of fecal

ARTICLE IN PRESS

Fig. 3. Marginal probability of fecal shedders of any Listeria mono-

cytogenes subtype plotted against the observed prevalence of fecal

shedders of any Listeria monocytogenes subtype with the pattern of silage

contamination and exposure to stress observed in the study herd over

three periods: first describing days 1–14, second describing days 26–29,

and third describing days 86–99. Note: there was no information on silage

contamination on days 25 and 85 and exposure to stress on days 0, 25 and

85. We assumed that cows were not exposed to stress on day 0, that they

were fed non-contaminated silage and were not exposed to stress on day

25, but were fed contaminated silage and exposed to stress on day 85. The

initial probability of cows in the shedding state was set to 0 for the first

and second periods, and to 0.5 for the third period.

R. Ivanek et al. / Journal of Theoretical Biology 245 (2007) 44–58

53

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shedders obtained on the study farm and the data obtained

on multiple other US dairy herds (Nightingale et al., 2004),

the study herd is not atypical with respect to the prevalence

of LM fecal shedders. As further data become available

they could easily be incorporated into the model to

improve generalizability of the model estimates.

The third assumption that we have made in the model

development is that any detected fecal shedding, contam-

ination of silage and exposure to stress are real, implying a

perfect sensitivity and specificity of tests used to detect

pathogens in feces (for brevity we will call it ‘‘fecal test’’)

and silage (we will call it ‘‘silage test’’) and a perfect

sensitivity and specificity of ‘‘perception’’ to detect cows’

exposure to stress (we will call it ‘‘stress test’’). A

consequence of this assumption is that we modeled the

detected rather than the true fecal shedding and its risk

factors. To model the true fecal shedding and its risk

factors would require knowledge about the sensitivity and

specificity of these tests, which are difficult to estimate

because they depend on many factors (such as detection

limit of the test and heterogeneity of animals). It is

reasonable to believe that the fecal, silage and stress tests

have perfect specificity, meaning that fecal and silage

samples with positive microbiological test results are truly

positive and cows’ exposure to antiparasitic treatment was

indeed stressful. While the sensitivity of tests used to detect

pathogens in fecal samples is often compromised, it seems

that the fecal test used to detect LM in feces of cows in the

study herd (Ho et al., 2006) has a reasonably good

sensitivity, particularly as the testing protocol utilized

enrichment of 10g fecal sample aliquots. The sensitivity of

this assay is further supported by isolation of LM from

feces of all tested cows (25) on 1 day of the study (day 87,

Ho et al., 2006), implying 100% (or close to 100%) test

sensitivity. In addition, fecal shedding was detected in 80%

and 84% of tested cows on two days of the study (days 29

and 88, see Ho et al., 2006), again implying a high level of

test sensitivity. While it thus seems likely that the sensitivity

of the fecal test in the study by Ho et al. (2006) was good, it

is also possible that the fecal test sensitivity varied from

day-to-day, fecal sample-to-sample, and/or animal-to-

animal with a probability distribution whose range

includes low but also high sensitivity levels to allow

identification of fecal shedding in all 25 cows tested on 1

day of the study. If the fecal test sensitivity is low, any

negative result obtained for detection of LM fecal shedding

at time t?1 and at time t could be positive with the

probability equal to 1-sensitivity. In our model, we are

interested in the probability that an animal sheds patho-

gens in feces at time t given the result of the test (its

shedding state) at time t?1. From this information for all

pairs of consecutive days we can construct a 2?2 table

showing the frequency of animals with respect to their

shedding state at times t?1 and t. It is immediately obvious

that entries in this 2?2 table could be different if the test

sensitivity is not perfect. Specifically, it is likely (not

certain!) that the true number of animals negative both at

times t?1 and t is lower but because of an imperfect test

sensitivity we were unable to detect pathogens in the feces

of some animals. Similarly, the true number of animals

positive both at times t?1 and t is likely (not certain!) to be

larger. However, the true number of animals in the rest of

the entries in this 2?2 table could both increase and

decrease. While intuitively and based on our analysis (not

shown here), this could cause both underestimation and

overestimation of the true stationary proportion of fecal

shedders, it is more likely to underestimate it. Likewise,

intuitively and based on our analysis (not shown here), an

imperfect fecal test sensitivity could underestimate and

overestimate the true duration of shedding and non-

shedding episodes, respectively.

While sensitivity of the fecal test may not be a problem,

the sensitivity of the silage and stress tests may be

questionable, leading to underestimation of the effect of

these risk factors on fecal shedding. Intuitively, and based

on our analysis (not shown here), this could cause

underestimation of the stationary proportion of fecal

shedders and the duration of shedding episodes and

overestimation of the duration of non-shedding episodes

in the presence of these risk factors.

To include possibly low sensitivity of any of the fecal,

silage or stress tests into the model one would have to

adjust the observed raw data before conducting statistical

analyses (univariate analysis and logistic regression).

However, because we do not know which of the negative

test results are actually false negatives, each negative test

would have to be given a chance of being positive with the

probability equal to 1-sensitivity, producing a different

result in each iteration. Thereafter, each iteration would

have to be analyzed in statistical analyses to estimate the

iteration specific parameter estimates for construction of

the iteration specific MC model, giving a distribution of

possible MC predictions. Incorporation of test sensitivity

into the developed MC model of LM fecal shedding would

allow modeling of the true rather than detected fecal

shedding and influence of risk factors on it. The problem,

however, is that it would significantly increase the

complexity of the model. More importantly, it would

depend on our very limited knowledge of the sensitivity of

these tests.

To validate the developed MC models, an ideal

approach would be to compare model estimates with a

comparable independent data set obtained through a

similar intensive longitudinal study of fecal shedding,

stress and feed contamination on other dairy farms.

However, although LM fecal shedding in dairy cattle has

long been recognized as an animal health and food safety

problem, the data on LM fecal shedding reported by Ho et

al. (2006), which we used for model parameterization,

represent the only data collected through an intensive

(daily) longitudinal sample collection. In the absence of

better validation data, the validity of the model was

examined by application of two less stringent approaches,

which we discuss below. The model estimates showed an

ARTICLE IN PRESS

R. Ivanek et al. / Journal of Theoretical Biology 245 (2007) 44–58

54

Page 12

encouraging agreement when compared to the data

observed in the study herd, meaning that the model was

able to capture and reproduce the dynamics of fecal

shedding observed in the data set and indicating a good

internal validity of the model. Comparing the model to the

data used for its parameterization may appear as blatant

circularity. However, Kendall et al. (1999) proposed that

this may not be as much of a problem as it appears. To

further validate the model, the predicted variability of

model estimates, varying between 2% and 92%, was

compared with the published data based on cross-sectional

sample collection in multiple US dairy herds, which

reported that the within-herd prevalence of cattle with

LM positive feces varies from a few percent to 100%

(Nightingale et al., 2004). While the reported variability

(Nightingale et al., 2004) is very similar to that predicted by

our model, the validity of comparison with data obtained

through a cross-sectional study design might be question-

able. The prevalence of fecal shedders estimated through

cross-sectional study design in a single herd represents a

point prevalence characteristic only for that point in time

and is a reflection of a specific combination of time-varying

covariates influencing that prevalence at that point in time.

However, the cross-sectional sample collection in multiple

herds captures different possible levels of within herd

prevalence of fecal shedders and the diversity of possible

combinations of time-varying covariates influencing it.

Therefore, the data reported by Nightingale et al. (2004)

not only represents the best available comparison data,

they also indicate that the within herd prevalence of fecal

shedders exhibits a high degree of variability. Our model

was able to capture and reproduce that variability while

still allowing analysis of the mechanisms (different

combinations of silage contamination and exposure to

stress) that cause it.

The results of our modeling work indicated that (i) the

prevalence of LM fecal shedders varies over time and can

be higher than 90%, (ii) LM subtypes exhibit different

dynamics of fecal shedding, (iii) the dynamics of LM fecal

shedding are highly associated with contamination of silage

and (iv) with cows’ exposure to stress, and (v) the

developed approach can be readily applied to study the

dynamics of fecal shedding in other pathogen–host–envir-

onment systems. These findings are discussed in detail

below.

The prevalence of LM fecal shedders varies over time and

can be higher than 90%: The predicted proportion of overall

and subtype specific fecal shedders varies considerably over

time (from 2% to 92%) depending on the pattern of time-

varying covariates. Such variability shows that any fecal

sample collection done less frequently than once a day is

not frequent enough because of a considerable day-to-day

variability in fecal shedding. While daily sample collection

is better than any less frequent sampling, particularly

monthly, we cannot say whether it is frequent enough

without further evidence of the within-day variability in the

pathogen fecal shedding. The predicted day-to-day varia-

bility in LM fecal shedding also points out that estimation

of the prevalence of fecal shedders based on data collected

through a cross-sectional study design in a single study

population (herd) has a limited value. Such an estimate

represents only point prevalence and by no means should

be used as a characteristic of the study population.

However, as discussed earlier, data on LM fecal shedding

collected through a cross-sectional study design in several

study populations (such as in the study by Nightingale et

al. (2004)), which are all exposed to different patterns of

time-varying covariates, should be able to reveal the true

variability of the prevalence of LM fecal shedders in the

general population.

The developed HMC models showed that a low

proportion of cows are likely to shed LM in feces even in

the absence of all considered risk factors (Table 3). This

implies that either (1) some relevant risk factor, such as

fecal shedding due to LM infection, was not considered, or

(2) silage was contaminated with LM but contamination

was not detected because of low silage test sensitivity

caused by uneven spread of LM in silage and presence of

LM in levels too low to be detected. LM infection and

prolonged shedding of a small proportion of cows in the

herd seem a reasonable explanation that may be supported

by a study (Fenlon et al., 1996) which reported shedding of

LM for at least 8 weeks after silage feeding had stopped.

Such prolonged shedding could be due to exposure of cows

to a source of LM other than contaminated silage but also

due to LM infection. The undetected silage contamination

hypothesis is supported by studies that reported that the

number and distribution of LM in silage varies consider-

ably. For example, in four out of six samples of silage

related to outbreaks of listeriosis, Wiedmann et al. (1996)

reported 5?103, 1?104, 1?105, and 1?108LM/g.

Another study reported LM numbers in spoiled silage of

0.9/g to 41.1?106cells/g (Fenlon et al., 1996).

LM subtypes exhibit different dynamics of fecal shedding:

The developed MC models indicated that the duration of

LM fecal shedding and the duration of non-shedding

periods vary for different LM subtypes. Fecal shedding of

ribotype 1058A could be particularly long; the estimated

mean duration was up to 21 days. The longest average

duration of shedding of ribotypes 1039E and 1042B is

much shorter, lasting only a few days. This subtype specific

variability in the dynamics of fecal shedding could in part

be due to intrinsic differences between LM subtypes.

However, it is also possible that it reflects the presence of

time-varying risk factors. Because LM from contaminated

silage could pass through a cow’s digestive tract and be

excreted through feces without causing LM infection

(Shepherd et al., 2000), the presence of specific LM

subtypes in silage may be responsible for subtype specific

dynamics of fecal shedding.

While estimated mean durations of time spent in the

shedding state were relatively short, some of the estimated

variances were large (Table 3), implying that most fecal

shedding episodes are short and a few very long. However,

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the estimated durations of fecal shedding could also be

explained by (i) a low fecal test sensitivity causing an

apparent short break in a long shedding period, and (ii) a

cow level heterogeneity displayed with long shedding of

some cows and short or absent shedding in other cows.

While the first explanation has already been discussed

under discussion of implications of modeling assumptions,

the second explanation does not seem likely based on the

similar pattern of fecal shedding exhibited by cows in the

study by Ho et al. (2006).

The dynamics of LM fecal shedding are highly associated

with contamination of silage: Silage contamination greatly

increases the probability of overall fecal shedding (all LM

subtypes) and fecal shedding of specific subtypes (ribotypes

1058A, 1039E and 1042B). This finding is in agreement

with multiple other studies. For example, in a study by

Fenlon et al. (1996), the same serotype and electrophoretic

type of LM was found in silage and in feces of cattle fed by

that silage. The same authors reported that 28.6–30.8% of

cattle started to shed LM after silage feeding started.

The equilibrium probability of overall fecal shedding and

fecal shedding of ribotype 1058A increased to a large extent

in the presence of contaminated silage, while the prob-

ability of fecal shedding of ribotypes 1039E and 1042B also

increased but far less (Table 3). Several explanations could

be proposed to explain the lower effect of silage

contamination on the probability of fecal shedding seen

in ribotypes 1039E and 1042B: (1) silage contamination

with these ribotypes is characterized by lower levels of LM

(below a detection limit of the silage test, implying a low

sensitivity of the silage test to detect these ribotypes) and a

smaller portion of silage contaminated, (2) ribotypes 1039E

and 1042B cause listeriosis infection and are subsequently

intermittently shed over a long period of time irrespective

of silage contamination status, and (3) several LM subtypes

are shed simultaneously but only one could be detected per

day because only one isolate was ribotyped per fecal

sample. The first explanation does not seem likely based on

the frequencies of silage samples contaminated with

different LM subtypes at low and high levels (Ho et al.,

2006); the ribotype 1048B was actually more commonly

present in silage at high levels (4 out of 5 positive samples).

The implications of low sensitivity of the silage test have

already been discussed. The second explanation is sup-

ported by isolation of ribotypes 1039E and 1042B in feces

on more days (8 and 13 days, respectively) than in silage

(3 and 4 days, respectively; Ho et al., 2006) indicating a

possibility of long term shedding after LM infection or

colonization. While this explanation seems realistic, it is

also possible that these ribotypes have been detected on

more days in cows’ feces compared to silage simply because

a larger number of fecal samples was examined per day

compared to fecal samples. The third explanation may

provide a reasonable clarification for the apparent lower

effect of contaminated silage on shedding of ribotypes

1039E and 1042B. The implications of low sensitivity of the

silage test have already been discussed. It has been reported

that feces could simultaneously be contaminated with more

than one subtype of LM (Ho et al., 2006). Ribotyping only

one isolate per fecal sample allowed identification of only

one LM subtype per sample of feces. A subtype that is

present in feces at the highest level is then the most likely to

be identified. While selecting a single isolate for typing is

not enough to capture the diversity of the LM population

in a single fecal sample, an unfortunate reality is that

isolating more than one isolate per sample would be cost-

prohibitive in a study that attempts to assess the dynamics

of fecal shedding in a population of animals. Nevertheless,

coexistence and competition of multiple ribotypes in a herd

and in an animal, either because of simultaneous infection

or passive shedding, may have a strong impact on the

dynamics of fecal shedding. However, this has been

omitted from our model because of lack of data.

The dynamics of LM fecal shedding are highly associated

with cows’ exposure to stress: The developed MC models

showed that exposure to stress can substantially increase

the probability of LM fecal shedding. This is in agreement

with a study that reported that stress caused by transport

increases shedding of LM in feces (Fenlon et al., 1996).

However, fecal shedding of different subtypes was not

equally affected by stress. Specifically, stress increased the

probability of overall LM fecal shedding (all subtypes) and

shedding of ribotype 1039E, but did not have any effect on

the probability of shedding of ribotypes 1058A and 1042B.

Several explanations could be proposed to elucidate why

stress did not have any effect on the probability of

shedding of ribotypes 1058A and 1042B: (1) the study

design did not have enough statistical power to detect that

association, (2) ingestion of ribotypes 1058A and 1042B

usually does not cause listeriosis, and shedding of these

subtypes represents only a passage through the cow’s

digestive tract, and (3) ingestion of ribotypes 1058A and

1042B could cause listeriosis in immunosuppressed animals

but feed was not contaminated with these subtypes on days

when cows were exposed to stress. The speculated

immunosuppressive effect of the antiparasitic treatment

that cows received during the study could be matched to

immunosuppression caused by stress (such as due to

transport) and certain physiological conditions (such as

pregnancy) which were reported to increase susceptibility

of an animal to LM infection (reviewed by Roberts and

Wiedmann, 2003). Therefore, in the presence of stress, the

probability of shedding should increase (such as it

increased for overall LM shedding and shedding of

ribotype 1039E) if cows become infected with LM from

silage. If after stress the cows do not start shedding, that is

because they did not become infected, because the

particular ribotypes cannot cause LM infection in cows

or, more likely, because there was no LM in the silage.

Indeed, the ribotypes 1058A and 1042B were not detected

in silage on days when cows were exposed to stress.

The developed approach can be readily applied to study the

dynamics of fecal shedding in other pathogen–host–environ-

ment systems: The applicability of the developed approach

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to study the dynamics of fecal shedding in other

pathogen–host–environment systems was demonstrated

using time series data describing fecal shedding of the

same pathogen in different species (LM in human) and

data on shedding of a different pathogen in the same

species (E. coli in cattle). Our estimate of the duration of an

episode of LM fecal shedding in healthy humans of 2 days

(mean; Var ¼ 3) is in agreement with the duration of

shedding of 1–4 days reported in a challenge study

(Angelakopoulos et al., 2002). Similarly, our estimate of

the prevalence of people that are likely to shed LM in their

feces (1.2%) is in agreement with studies that reported the

prevalence of LM fecal shedders to be 2–6% (Rocourt and

Cossart, 1997), 1.8–9% (Ralovich, 1984), and 0.8% (Grif

et al., 2001) among healthy people. The agreement between

the estimates from the model of LM fecal shedding in

humans and published data further confirms the validity of

the developed MC modeling approach to study the

dynamics of pathogen fecal shedding.

5. Conclusion

In an effort to develop an analytical tool for under-

standing and predicting the dynamics of pathogen fecal

shedding, we proposed an MC analysis. While MC

modeling has been extensively applied in many settings,

including transmission of infectious diseases (e.g. Chowell

et al., 2004; Lesnoff et al., 2004; Penne and Haese, 1999) we

are not aware of its application in modeling of the

dynamics of fecal shedding. The developed MC models

allowed estimation of the day-to-day and equilibrium

proportions of dairy cattle likely to shed LM in feces, and

the duration and frequency of LM fecal shedding depend-

ing on the presence of time-varying covariates (here,

contamination of feed and exposure to stress). While it

has been accepted that the nature of LM shedding is

intermittent, to our knowledge, there are no reported data

on the duration of shedding and non-shedding episodes of

LM in cattle or in other animals including man. Estimated

durations of LM fecal shedding and non-shedding episodes

may be useful in development of a mathematical model of

LM transmission, such as proposed by Ivanek et al. (2006).

The developed MC modeling introduced structure into the

dynamics of pathogen fecal shedding and as such will be

useful to modelers and scientists studying pathogen

transmission through fecal shedding, with the ultimate

goal of understanding and controlling diseases that spread

through fecal shedding.

Acknowledgments

This research was supported in part by a USDA Special

Research Grant (2003-34459-12999) and in part by the

Cornell University Agricultural Experiment Station federal

formula funds, Project No. NYC-143451 received from

Cooperative State Research, Education, and Extension

Service, US Department of Agriculture. Any opinions,

findings, conclusions, or recommendations expressed in

this publication are those of the author(s) and do not

necessarily reflect the view of the US Department of

Agriculture.

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